Bulletin of Volcanology

, Volume 73, Issue 1, pp 55–72 | Cite as

Spatio-temporal hazard estimation in the Auckland Volcanic Field, New Zealand, with a new event-order model

Research Article

Abstract

The Auckland Volcanic Field (AVF) with 49 eruptive centres in the last c. 250 ka presents many challenges to our understanding of distributed volcanic field construction and evolution. We re-examine the age constraints within the AVF and perform a correlation exercise matching the well-dated record of tephras from cores distributed throughout the field to the most likely source volcanoes, using thickness and location information and a simple attenuation model. Combining this augmented age information with known stratigraphic constraints, we produce a new age-order algorithm for the field, with errors incorporated using a Monte Carlo procedure. Analysis of the new age model discounts earlier appreciations of spatio-temporal clustering in the AVF. Instead the spatial and temporal aspects appear independent; hence the location of the last eruption provides no information about the next location. The temporal hazard intensity in the field has been highly variable, with over 63% of its centres formed in a high-intensity period between 40 and 20 ka. Another, smaller, high-intensity period may have occurred at the field onset, while the latest event, at 504 ± 5 years B.P., erupted 50% of the entire field’s volume. This emphasises the lack of steady-state behaviour that characterises the AVF, which may also be the case in longer-lived fields with a lower dating resolution. Spatial hazard intensity in the AVF under the new age model shows a strong NE-SW structural control of volcanism that may reflect deep-seated crustal or subduction zone processes and matches the orientation of the Taupo Volcanic Zone to the south.

Keywords

Monogenetic volcanism Basaltic volcanic fields Probabilistic hazard Tephrostratigraphy 

Introduction

The estimation of hazard in regions of distributed small-volume basaltic volcanism (monogenetic volcanic fields) is fraught with the challenges of understanding both the past frequency of eruption events and the order of events in relation to their spatial distribution (Connor and Hill 1995; Condit and Connor 1996; Conway et al. 1998). Such areas of volcanism are common to most parts of the Earth, and occur in a range of tectonic environments (Takada 1994; Valentine and Gregg 2008). Since the next eruption of a monogenetic field will most probably occur in a new location, any hazard forecast must consider not only the probability of when eruptive activity will resume, but also where. In urbanised volcanic fields, such as the 1.4 million population Auckland City, New Zealand, or locations with a high amenity/infrastructure value, such as the proposed Yucca Mountain Nuclear Waste Repository in Nevada, the question of where new eruptions may occur is especially critical for socio-economic risk assessment (Connor et al. 2000; Magill et al. 2005). These spatio-temporal relationships are also highly important to the understanding of the fundamental structure of volcanic fields and the processes behind distributed volcanism (Connor and Conway 2000).

Key hindrances to developing a hazard forecast for monogenetic field volcanism include establishing the ages and frequency of past eruptions. This is especially true in Late Quaternary volcanic systems, such as the AVF, where a combination of young lavas with low-K composition (1.3–1.7% K2O; Mochizuki et al. 2004) hinders application of Ar–Ar dating for the youngest eruptive products, and excess Ar in phenocrysts hampers conventional K–Ar dating (McDougall et al. 1969). Eruptions of field volcanoes are also often small, with very restricted deposits that frequently fail to overlap those of neighbouring centres. This reduces the chances of establishing a robust stratigraphic ordering. In addition, many volcanic fields, such as the AVF, display either little or conflicting evidence for structural control of vent locations (Spörli and Eastwood 1997; von Veh and Nemeth 2009), and a typical absence of obvious age:location trends. Because of these constraints the most common spatio-temporal hazard assessments for distributed volcanic systems rely on defining a vent clustering model (Magill et al. 2005), a structural/age model (Connor and Hill 1995; Cronin et al. 2001), or a geomorphology-based age model (Hasenaka and Carmichael 1985), or simply averaging the hazard over the field area.

In the Late Quaternary Auckland Volcanic Field (AVF; Fig. 1), 49 eruption centres occur over a 360 km2 area, with well constrained location and volume information (Allen and Smith 1994). Age-dating in the field has, however, been extremely challenging, with one recent study (Lindsay and Leonard 2009) concluding that only nine of the eruptive centres can be dated with a high degree of reliability. Despite this fact, there is a large amount of other age-related information for the AVF that can be combined and used to build a robust age model and order events/locations. This includes dated tephras contained within sediment cores collected from a number of volcanic depressions in the region (e.g., Molloy et al. 2009). Geochemical correlation of these tephras to their source vents is made difficult by there being a low overall range of magma compositional variability in the field (Smith et al. 2009). In addition, individual eruptive centres/events may express a high degree of variability within the overall field range (cf. Smith et al. 2008) making unique correlations unlikely. A further limitation is the lack of available chemical datasets appropriate for correlation studies. The volcanic centres are represented dominantly by whole-rock analyses from lava flows or coherent bomb clasts (Smith et al. 2009), whereas the fine particle size in most tephra cores precludes representative whole-rock analysis and typically only glass chemistry is available (Molloy et al. 2009). Understanding these limitations, but using geochemistry as an aid where appropriate, we build a new age model for the AVF using a combination of (1) the known age constraints from the volcanic constructs from a range of sources; (2) all known physical stratigraphic relationships of volcanic products; and (3) a probabilistic correlation of dated tephra layers from sediment cores to their most likely volcanic source vents. This model shows major improvements in consistency over past attempts and, based on it, we present here a major re-evaluation of probabilistic hazard models for the field, along with new insights into its temporal and structural history.
Fig. 1

Volcanic geological map of the Auckland Volcanic field, North Island, New Zealand, adapted from Kermode (1992), showing named volcanic centres and the major lithologies of eruptives. Coring sites for volcanic ash layers (Table 1) are located at: Lake Pupuke, Hopua, Orakei Basin, Onepoto Basin and Pukaki (indicated by stars)

Past dating of eruption centres in the AVF

Attempts at precisely dating AVF volcanoes began with the early application of 14C methods (Fergusson and Rafter 1959) and have been carried out using a variety of radiometric techniques ever since, with varying degrees of success. Searle (1961) provided an early age-model, combining a few radiometric ages with stratigraphic and geomorphic constraints on some of the volcanoes. In the detailed mapping work of a number of workers, including Firth (1930), Searle (1959a, b, 1962, 1964, 1965) and later Kermode et al. (1992), field stratigraphic relationships are presented that assist in ordering the eruption events, even if not directly indicating their ages.

Allen and Smith (1994) summarized published and unpublished ages for the AVF in a comprehensive way, and this information has been recently updated by Lindsay and Leonard (2009). Both studies recognise the poor correlation between age estimates from different methods and conclude that few events are well constrained; the latter study classifying up to 35 eruption centres as without any form of reliable date. Part of the problem was recognised by McDougall et al. (1969) in that K–Ar ages from the field were considerably older than those provided by other means, due to excess Ar in the basaltic rocks. This was improved on by Mochizuki et al. (2004), and more recently still further with the application of incremental Ar–Ar dating (Cassata et al. 2008). Except for the most recently obtained of these ages, however, estimates vary greatly for the eruptions of single centres. The most consistent age determinations are from the 14C method, especially where the organic material analysed was trapped (and charcoalised) beneath volcanic products. Notably, such 14C ages are concordant with all of the newer Ar–Ar ages from the field.

In addition to dating studies, Rout et al. (1993) recognised that several of the volcanoes had products with similar magnetic field properties, implying coeval eruption. Cassidy (2006) went on to demonstrate that five eruptions (Crater Hill, Mt Richmond, Puketutu, Taylor Hill and Wiri Mountain) may have occurred during a period of no more than 100 years during the Mono Lake excursion (age ∼29 ka). Cassata et al. (2008) also identified the Lashamp geomagnetic excursion and provided more precise ages for the eruptions during these two excursions. These ages appear to be consistent, within the error ranges, with the available 14C ages.

Alongside the attempts to date individual eruption centres, an arguably even more reliable source of age information in the AVF has been derived from coring sediments within maar depressions throughout the field (Sandiford et al. 2001; Shane and Hoverd 2002; Molloy et al. 2009). There is, however, an absence of reliable geochemical information to link these determinations to particular eruptions. Hence, methods for combining core tephra dates (Turner et al. 2009) are not applicable. Shane and Hoverd (2002) combine a tephra record from Onepoto Basin with one from the Pukaki crater (Sandiford et al. 2001) to constrain the ages of 15 AVF events. Molloy et al. (2009) extended this to include records from Lake Pupuke, the Hopua Basin, and the Orakei Basin, detecting a total of 24 AVF events. Unfortunately, there is no evidence as to which volcano they correspond.

Constructing a new age model for the AVF

The cores collected in five locations (Fig. 1; Table 1) contain 24 tephras from the AVF (Molloy et al. 2009). These cores provide estimated dates that are constrained by deposition rates, 14C determinations and tephrochronology of distal tephras from other North Island volcanoes. Geochemical correlation of the tephras to eruption centres has not been attempted because the inter-shard variation for the basaltic AVF tephras is high, precluding unique correlations to source cones (Shane and Hoverd 2002). Based on the unambiguous ages of the two most recent tephras, AV24 can be correlated to Rangitoto, and AV23 to Mt Wellington. To correlate the others to known sources we can make use of the known thickness, location, and stratigraphy, and in some cases the (14C) age, which can be combined to assemble a coherent correlation model. Consider the relation
$$ r = - 1.85{V^{1/3}} + \exp \left[ {(8.67 + 1.13\log V - \log T)/2.38} \right] $$
(1)
from Rhoades et al. (2002), where V is erupted volume of tephra (km3), T is the deposited tephra thickness (cm), and r is the distance from the centre of deposit (km). We note that Eq. 1 is obtained by analysis of eruption volumes as small as 107 m3, which is less than the average pyroclastic volume from events in the AVF. Although the model has corrections for wind effects, these are high-level winds, which are consistently from the West in New Zealand. However, dispersal from the 1–3 km altitude plumes assumed for the AVF is dominated by low-level winds which, judging by recent data (Reid 1980) exhibit none of the consistency of higher level winds. Although any given eruption will have occurred in a windfield, we have no information regarding the wind direction or strength, and so the least biased solution is to ignore wind correction terms, and assume the vents lie at the centres of deposits. We need simply to keep in mind that there will be wind errors, and that the derived correlations need to be tested further by geochemical or mineralogical methods when data become available.
Table 1

Thickness and estimated age of AVF tephra in five cores, summarised from published data of Sandiford et al. (2001); Shane and Hoverd (2002); Molloy et al. (2009). Age errors are 1σ

ID

Thickness (mm)

Age (ka)

Pupuke

Onepoto

Orakei

Hopua

Pukaki

Pupuke

Onepoto

Orakei

Hopua

Pukaki

AV24

22

    

0.8 ± 0.1

    

AV23

   

3

    

10.0 ± 0.7

 

AV22

    

1

    

15.1 ± 0.7

AV21

   

290

3

   

20.0 ± 0.7

19.7 ± 0.7

AV20

   

235

3

   

20.5 ± 0.7

19.7 ± 0.7

AV19

    

1

    

24.4 ± 0.8

AV18

  

8

40

0.5

  

24.5 ± 0.8

26.6 ± 0.8

24.7 ± 0.8

AV17

  

5

    

24.7 ± 0.8

  

AV16

    

50

    

25.5 ± 0.8

AV15

  

12

    

26.7 ± 0.9

  

AV14

  

340

    

26.8 ± 0.9

  

AV13

  

740

    

26.9 ± 0.9

  

AV12

7

12

410

335

2

28.8 ± 0.9

28.7 ± 0.9

30.6 ± 0.8

29.0 ± 0.8

28.1 ± 0.9

AV11

    

2

    

30.5 ± 0.9

AV10

3

    

30.6 ± 0.9

    

AV9

6

 

400

 

850

30.8 ± 0.9

 

31.5 ± 0.8

  

AV8

20

2

45

 

250

31.1 ± 0.9

30.9 ± 0.9

31.5 ± 0.8

 

32.1 ± 0.9

AV7

2

4

20

 

2

32.0 ± 0.9

30.9 ± 0.9

31.5 ± 0.8

 

32.3 ± 0.9

AV6

 

8

  

555

 

31.9 ± 0.9

  

32.6 ± 0.9

AV5

  

110

    

34.8 ± 0.8

  

AV4

15

12

35

  

34.4 ± 0.9

35.0 ± 0.9

35.4 ± 0.8

  

AV3

  

120

    

54.7 ± 4.6

  

AV2

  

510

    

68.3 ± 6.4

  

AV1

 

190

40

   

75.0 ± 5.4

83.1 ± 5.4

  
In order to validate the relation (Eq. 1) for applicability to the AVF, including the insensitivity to wind effects, we will follow the approach of Magill et al. (2006), and examine the 1977 Ukinrek Maar eruptions, Alaska. These eruptions produced two maars with an estimated erupted volume of 0.026 km3 (Kienle et al. 1980). Although the windfield (Fig. 4a of Self et al. 1980) exhibits a predominantly NNW and SE direction, the actual isopachs are almost circular (Fig. 1 of Self et al. 1980), indicating that at low levels, the wind correction may be neglected. Hence Fig. 2 shows the measured tephra thicknesses from Self et al. (1980), with radius distances measured from a point equidistant between the two vents, and the predicted thickness-distance relation from Eq. 1. This shows that the relation (Eq. 1) is reliable (cf. Magill et al. 2006), with possibly a tendency to under-estimate the distance for large volumes, and to over-estimate the distances for smaller volumes. However, the latter effect acts against the expected error due to omission of the wind correction, which we will bear in mind in our subsequent analysis.
Fig. 2

Tephra thicknesses at distances from a point midway between the two vents of the 1977 Ukinrek Maars eruption, and the predicted relation (solid line) from Rhoades et al. (2002)

Eruption volumes for each centre in the AVF were calculated by Allen and Smith (1994) using a number of methods:
  1. 1.

    If the area of the explosion crater was known, tuff volume was calculated as \( (1/3)\pi {r^2}h - \left( {\pi r_c^2{h_t} + (1/3)\pi r_c^2\left( {h - {h_t}} \right)} \right) \), where r is the radius of the tuff deposits plus the explosion crater, rc the radius of the explosion crater, h the estimated (by extrapolation) height of deposits at the centre of the crater, and ht is the height of the tuff ring.

     
  2. 2.

    Calculating the volume of country rock removed in the explosion crater from geophysical information, adding 30% for the magmatic content.

     
  3. 3.

    Using method 1 with the addition of a tuff base in cases where erosion eliminated distal tuff deposits.

     
  4. 4.

    Where limited data was available, using the formula \( (1/3)\pi {r^2}h \).

     

The resulting volume estimates range from ∼10−4 km3 (Pukeiti, Hampton Park and Purchas Hill) to 0.1014 km3 (Three Kings), with a mean of 0.0137 km3.

For each core listed in Table 1, and using the pyroclastic eruption volumes calculated by Allen and Smith (1994), the relationship (Eq. 1) can be used to construct graphs showing the thickness to be expected at each core site for each other eruptive centre in the field. An example for Pupuke is shown in Fig. 3. The multiple tephra thicknesses of hundreds of mm mean that many of them must have been from sources within a few km, because there have been few events large enough to produce such thicknesses at distance, even allowing for the (unknowable) ambient winds and the tendency noted above for possible underestimation of distance in Eq. 1. Magill and Blong (2005) note that the median base surge area in the AVF is 2.63 km2 and, at Ukinrek, thicknesses of 35–40 cm are found at distances of 1–2 km from the vents (Self et al. 1980). Another potential effect is possible over-thickening of some of the tephras within basins due to rainfall-induced remobilisation of tephra from the surrounding landscape. Secondly, events that appear to be recorded at Hopua and/or Pukaki in the south (Fig. 1), as well as Pupuke and/or Onepoto in the north, are likely to have been both large and central. Using these obvious starting points, we tested many various ways of assigning events to one of the 24 tephra layers. Dates directly from eruption centres (especially 14C ages) that are considered reliable (Table 2) must of course match those tephra dates in Table 1 for any given assignment. In addition stratigraphic constraints (Table 2) must be respected. Further, all tephra layers have to be assigned. We also apply the assumption that an event cannot be identified with a tephra at a distal site, while being absent at a more proximal one on the same orientation. This effectively excludes the geological possibility of tephra erosion, but the depositional sites were lakes and most sequences show continuous sedimentation (e.g., Molloy et al. 2009). Hence, the chances of missing a proximal tephra in the record are very low. Combining these perspectives, the constraints on event identification become very tight, even allowing for a degree of subjectivity derived from the unknown wind effects.
Fig. 3

Tephra attenuation curves for Pupuke core applying the relationship proposed by Rhoades et al. (2002). The numbers indicate the volume and distance of the vent, numbered according to the order in Allen and Smith (1994)—see Table 2. The curves show the combination of distance and volume expected to produce the observed tephra thickness at Pupuke. The dotted curve is for a thickness of 0.5 mm, which is considered the detection threshold, and solid curves proceed to the left, indicating thicknesses of 2, 3, 6, 7, 15, 20 and 22 mm

Table 2

New age model for the AVF, combining current known age constraints and geomorphology of the AVF volcanic structures and probabilistic correlation to tephra layers (as described in “Constructing a new age model for the AVF”), compared to the Allen and Smith (1994) assigned ordering of events. 14C determinations are here considered to be the most reliable eruptive centre ages <30 ka B.P. and are deferred to if contrasting with results from other radiometric methods. Where multiple consistent 14C ages are reported, the combined result and error is listed. Whole-rock K–Ar determinations are considered unreliable due to the presence of excess Ar in phenocryst phases (McDougall et al. 1969) and are hence excluded. Shaded units lack age constraints independent of tephra correlation

aReported in calibrated years B.P. for 14C age determinations (CalPal Online; www.calpal.de) or tephrostratigraphic correlations based on radiocarbon chronology. Other types of age determinations are italicised. Bold italic indicates age determinations obtained from the tephra matching process

bGeomorphologically and stratigraphically based relative-age estimations are sourced from Firth (1930), Searle (1959a, b, 1961, 1962, 1964, 1965), Kermode et al. (1992), Allen and Smith (1994) and Affleck et al. (2001)

cGrenfell and Kenny (1995)

dTephrostratigraphy, and some 14C dates, Lindsay and Leonard (2009)

eFergusson and Rafter (1959), Grant-Taylor and Rafter (1963, 1971)

fShane and Sandiford (2003); whole lapilli Ar–Ar determination

gMolloy et al. (2009); tephrostratigraphy

hCassata et al. (2008); Ar–Ar date no excess Ar reported

iPolach et al. (1969); McDougall et al. (1969)

jSandiford et al. (2002)

kWood (1991), thermoluminescence age on plagioclase crystals

lHayward (2008)

mNewnham et al. (1999, 2007); tephrostratigraphy

nShane (2005)

oSibson (1968)

pSameshima (1990)

qCombination of three TL ages from Adams (1986), Phillips (1989) and Wood (1991)

rCombination of six 14C ages from Eade (2009)

sEast and George (2003) radiocarbon age

tHorrocks et al. (2005b)

uCalibrated ages, combinations of several radiocarbon ages using OxCal 4.1 (c14.arch.ox.ac.uk; curve ShCal04), as reported in Lindsay and Leonard (2009)

vTephra subscripts indicate the coring sites where they are described Onepoto (a), Pupuke (b), Pukaki (c), Orakei (d), and Hopua (e), see Table 1

The depositional ages of tephras in Table 1 were estimated (Molloy et al. 2009) by linear interpolation, and occasionally extrapolation, between marker tephras and 14C dated horizons. As Molloy et al. (2009) provide no details of the precision of individual measurements within each core; this was estimated, for AV4-AV24, using a Monte Carlo procedure. In this procedure each control age was simulated from a normal distribution with standard deviation given by Lowe et al. (2008) or Molloy et al. (2009), and the interpolated ages recalculated. The estimated (1σ) error was then taken as the standard deviation from a sample of 1000 repetitions. However, there remains the error derived from the assumption of a constant sedimentation rate. We note that the estimated ages for those tephras considered by Molloy et al. (2009) as being present in at least three cores have standard deviations ranging from 0.5 ka to 1.2 ka. Hence we will use 0.7 ka, the mean of these, as a proxy for the error inherent in the linear approximation, which is combined with the precision error by adding the variances to obtain error estimates for each age. For AV1-AV3, the error was estimated as the sample standard deviations for the ages of the Eg10, Eg4 and Eg2 tephras (see Molloy et al. 2009) for AV3, AV2, and AV1 respectively. This last gives a somewhat larger estimated error than is claimed in Molloy et al. (2009), but as noted in the discussion in Molloy et al. (2009), these three ages are obtained by extrapolation rather than interpolation.

This does not, however, address the issue of systematic bias within a core, as exemplified by the differing age estimates in different cores for the same tephra. Hence, after a tephra has been correlated to a given eruption centre, that centre’s (14C) age is assigned to the tephra, and in a stepwise fashion used to ‘calibrate’ other ages estimated from the same core. Thus, and especially as we sometimes disagree with Molloy et al. (2009) about the common origin of different tephra deposits, we do not use the combined tephra ages across cores. Rather, we assume that each core has a systematic error, and estimate that via our correlation procedure. The assumed precision is then obtained by adding the variances in the assigned age and in the correlative age. In the case of multiple estimated ages from different cores, they are combined using the procedure of Ward and Wilson (1978). Turner et al. (2008b, 2009), proposed a different procedure for the tephra record at Mt Taranaki, using splines to estimate the sedimentation rate, and matching different cores via age distribution and principal component analysis of the tephra geochemistry.

Using the above methods and constraints, the AVF tephra correlation incorporates known age constraints, stratigraphic and geological criteria, with the final assignment of tephras to vents listed in Table 2. Note that some tephra layers correlated by Molloy et al. (2009) between sites in Table 1 were considered to be from different eruption centres, because of the wide separation in space or time, and conditional on previous assignments. The process of correlation proceeded as follows:
  1. 1.

    Exclude Onepoto (248 ka), Pupuke (200 ka), Orakei (>AV1), and Pukekiwiriki (> 125 ka) from consideration, because they are all older than the tephra sedimentary records.

     
  2. 2.

    Assign AV24 to Rangitoto, AV23 to Mt Wellington 1.

     
  3. 3.

    Assign AV12 to Three Kings, because it is present in all cores. Most likely this was sourced from a central location, and it was a large event. These criteria are most closely matched by Three Kings, which has also a well-fitting eruption centre age.

     
  4. 4.

    Assign AV13, 14 to Mt Eden and Mt Hobson, respectively. They are large tephras in Orakei, hence probably sourced from closely located vents, especially as they are not recorded elsewhere. Mt Eden and Mt Hobson have the desired stratigraphy, age, and location. Calibration of Orakei core ages by the Mt Eden 14C date gives an age of 28.3 ± 1.0 ka for Mt Hobson.

     
  5. 5.

    Assign AV4 to One Tree Hill, because it occurs in all northern cores and is the best combination of size, location and stratigraphy. This provides a calibrated age of 37.2 ± 0.8 ka from the Hopua assigned age (see 14. below).

     
  6. 6.

    Assign AV11 to Wiri Mountain, since it is in only the Pukaki core and has the right age.

     
  7. 7.

    Assign AV9 to Panmure Basin, due to it being found in Pupuke, Orakei and Pukaki cores. Panmure Basin has the optimum location, size and eruptive style to have generated the observed tephra deposits, and is also approximately the right age.

     
  8. 8.

    Assign AV6 [in the Pukaki core] to the nearby Crater Hill, because it is a very large tephra with the right age. The ‘Skip-over rule’ means that AV6 [in the Onepoto core] must represent a different event.

     
  9. 9.

    Assign AV18, 20 and 21 to Mangere Lagoon, Mt Mangere and Mt Smart. This is the only feasible assignment that does not result in contradictions with later assignments; they are also consistent with locations and eruption volumes. Calibrated age for Mt Smart becomes 21.7 ± 0.6 ka from the Mt Mangere 14C age. The calibrated age for Mangere Lagoon becomes 27.0 ± 0.5 ka from the Mt Eden and Mt Mangere 14C ages.

     
  10. 10.

    Assign AV8 [in the Orakei and Pukaki cores] to Puketutu, due to it having the correct age, location and size. AV8 [in the Pupuke and Onepoto cores] appears to be a different event.

     
  11. 11.

    Assign AV16 to Waitomokia, because it is the only likely location remaining. The calibrated age estimate of this event is then 27.8 ± 1.0 ka from the Mangere Lagoon assigned age.

     
  12. 12.

    Assign AV8 [in the Pupuke and Onepoto cores] to North Head, applying the skip-over rule. This provides a calibrated age of 32.7 ± 1.0 ka from the Panmure Basin 14C age.

     
  13. 13.

    Assign AV10 to Mt Victoria, applying the skip-over rule. This implies a calibrated age of 32.2 ± 1.0 ka from the Panmure Basin 14C age.

     
  14. 14.

    Assign AV7 to Hopua, due to its central location, size and minimum age. The calibrated age will hence be 33.7 ± 1.0 ka from the Crater Hill 14C age.

     
  15. 15.

    Assign AV2 to Domain, as the distance–thickness relationship at Orakei is consistent with the eruption from Domain, and the estimated tephra age exceeds the age constraint on Domain of >60 ka. Age is estimated at 68.3 ± 6.4 ka.

     
  16. 16.

    Assign AV5 to Motokorea, because it is the only event with a thickness–distance relationship matching that of AV5 left unassigned, and Motokorea’s low SiO2 content matches the glass chemistry of AV5 rather than AV3 (Molloy et al. 2009, Fig. 7a). This results in a calibrated age of 36.6 ± 1.1 ka from the One Tree Hill age assignment.

     
  17. 17.

    Assign AV3 to Mt St John, which is the best possibility remaining, resulting in an estimated age of 54.7 ± 4.6 ka.

     
  18. 18.

    Assign AV1 [in the Orakei core] to Pukaki, with an estimated age of 83.1 ± 5.4 ka.

     
  19. 19.

    Assign AV6 [in the Onepoto core] to Mt Albert (the only other possibility is Albert Park, where the high degree of erosion and weathering is inconsistent with the ∼34 ka age estimated for AV6). This results in a calibrated age of 34.7 ± 1.4 ka from the Hopua assigned age.

     
  20. 20.

    Assign AV15 and 17 to Pigeon Mountain and Robertson Hill, respectively. Any alternative remaining candidate source vents are either too old or located too far south. This results in a calibrated age of 28.2 ± 1.0 ka for Pigeon Mountain from the Mt Eden 14C age and 27.2 ± 1.0 ka for Robertson Hill from the Mangere Lagoon assigned age.

     
  21. 21.

    Assign AV22 to Matakarua, resulting in an estimated age of 15.1 ± 0.7 ka.

     
  22. 22.

    Assign AV19 to Otara Hill, giving a calibrated age of 26.7 ± 1.0 ka from the Mangere Lagoon age assignment.

     
  23. 23.

    The only remaining source compatible with AV1 [in the Onepoto core] (190 mm) is Tank Farm. As the stratigraphy here is uncertain (Ian Smith, pers. comm.), we will accept this, with an assigned age of 75.0 ± 5.4 ka. The only other possibility is a conjectured second phase of Pupuke (Ian Smith, pers. comm.).

     

The augmented age information from the assignment routine is shown in Table 2.

From this process we now have assigned ages for 39 out of 51 events (49 vents, with multiple eruptions at Mt Wellington and Rangitoto). Of the remaining 12 events, Taylor’s Hill is considered equivalent in age to the Crater Hill group, based on paleomagnetism (Cassata et al. 2008) and Mt Cambria may be of very similar age to its partly coalesced neighbour Mt Victoria. Stratigraphic relationships can be used to bound the ages of five further centres (Mt Roskill, Pukeiti, Otuataua, Styaks Swamp and Little Rangitoto), assuming that the only centre younger than Mt Wellington is Rangitoto. Stratigraphic relationships provide lower bounding ages for another four centres (St Heliers, Pukekiwiriki, Te Pouhawaiki and Orakei Basin). Albert Park has no useable stratigraphic correlations and is only assumed to be one of the earliest eruptions in the field, based on its degree of weathering and subdued geomorphology (Searle 1962).

Description of AVF temporal behaviour

The result of “Constructing a new age model for the AVF” is an age or event-order determination with quantifiable unknowns for the AVF. Based on this we can examine the behaviour of the field by repeated Monte Carlo simulation of the ages, analysis of the simulated ages, and calculation of the mean and variance of the results. Ages were assigned via the following simulation algorithm:
  1. 1.

    The 39 events with estimated ages and errors were assigned an age from a normal distribution with the given mean and error.

     
  2. 2.

    Mt Cambria is sampled from the same distribution as Mt Victoria, and Taylor’s Hill from a normal distribution with average and error given by the sampled ages of Mt Richmond, Wiri Mountain, Puketutu and Crater Hill.

     
  3. 3.

    If the sampled ages violate the stratigraphy, new random values are drawn for all the affected events. Note that the tephra assignments generate additional implied stratigraphy within each core.

     
  4. 4.

    The remaining 11 events are finally assigned an age that is uniformly distributed between any upper and lower stratigraphic bounds, with an assumed upper bound of 270 ka or the age of Onepoto Basin, whichever is greater. The age of St Heliers is assumed to be at least 90 ka, and Albert Park is assumed to be older than St Heliers (although this cannot currently be verified with certainty).

     
The results of 1000 Monte Carlo simulations are shown in Table 3, ordered by mean age. Notable features are the decomposition into six ‘early’ events (Onepoto Basin, Albert Park, Lake Pupuke, Pukekiwiriki, St Heliers and Orakei Basin), followed by (in order) the ‘intermediate’ events of Pukaki, Tank Farm, Domain and Mt St John. Following there is a period, from around 40 ka to 20 ka, probably beginning with the Maungataketake eruption, containing 31 or more events. A period from 20 ka to 10 ka with a few poorly-defined events is followed by the eruptions of Purchas Hill, Mt Wellington and Rangitoto as the most recent events. Unfortunately, the Little Rangitoto event remains almost completely unconstrained, and Te Pouhawaiki is little better.
Table 3

Monte Carlo ages and ordering from 1,000 simulations of the new AVF age-order model with constraints described in “Description of AVF temporal behaviour”. Order(AS94) indicates the order within Allen and Smith (1994)

Name

Mean age (ka)

Age error (ka)

Order (AS94)

Min order

Max order

Onepoto Basin

246.8

28.8

4

1

8

Albert Park

227.6

41.2

2

1

7

Lake Pupuke

200.2

7.3

6

1

8

Pukekiwiriki

195.1

44.2

16

1

8

St Heliers

182.3

53.0

3

2

8

Orakei Basin

178.5

55.8

45

1

7

Te Pouhawaiki

152.0

70.0

40

1

33

Little Rangitoto

92.3

57.7

44

2

46

Pukaki

83.4

5.5

29

7

11

Tank Farm

75.2

5.5

5

6

12

Domain

69.3

5.3

1

7

12

Mt St John

54.7

4.5

42

9

14

Maungataketake

41.4

0.4

18

11

15

One Tree Hill

37.4

0.7

37

12

17

Motukorea

36.1

0.9

48

13

20

Mt Albert

35.2

0.9

10

14

21

Kohuora

35.1

1.0

23

13

21

Crater Hill

34.1

0.3

24

15

22

Hopua Basin

33.6

0.4

39

18

23

Puketutu

33.0

0.5

26

19

25

North Head

32.9

0.4

9

19

26

Hampton Park

32.4

4.8

34

12

41

Panmure Basin

32.3

0.3

12

21

27

Taylors Hill

32.0

1.9

28

13

39

Mt Roskill

32.0

1.9

11

16

34

Ash Hill

31.8

0.2

22

21

30

McLennan Hills

31.7

0.2

13

22

30

Mt Victoria

31.5

0.6

8

22

33

Mt Cambria

31.4

1.2

7

16

38

Wiri Mountain

30.9

0.4

25

24

32

Mt Richmond

29.9

0.6

14

25

35

Three Kings

28.8

0.3

38

29

35

Mt Eden

28.4

0.3

41

31

36

Waitomokia

28.1

0.8

17

27

39

Mt Hobson

28.1

0.3

43

32

37

Pigeon Mt

27.7

0.4

27

33

39

Robertson Hill

27.2

0.4

15

34

40

Mangere Lagoon

26.7

0.4

35

36

41

Otara Hill

25.9

0.7

33

38

42

Pukeiti

22.3

4.2

19

30

46

Mt Mangere

22.1

0.4

36

39

44

Green Hill

21.9

7.1

32

12

46

Mt Smart

21.4

0.5

30

40

45

Otuataua

16.4

4.3

20

36

47

Styaks Swamp

15.9

5.1

31

14

47

Matakarua

15.1

0.7

21

42

46

Purchas Hill

10.8

0.1

46

45

47

Mt Wellington

10.5

0.1

47

48

48

Mt Wellington 2

10.1

0.1

50

49

49

Rangitoto

0.6

0.0

49

50

50

Rangitoto 2

0.5

0.0

51

51

51

Hazard models for the AVF

The usual object of stochastic modelling in monogenetic fields is a spatio-temporal estimate of the likely timing and location of the next event (Connor and Hill 1995; Conway et al. 1998; Cronin et al. 2001; Martin et al. 2004). Eruptive volume is not generally considered, except when the spatial dimension is heavily discretized (Bebbington 2008). The hazard model we attempt to construct here is a probabilistic model with point process intensity λ(x,t) such that the probability of an event in the time interval (t, tt) in an area of size ΔA (which includes x) is approximately \( \lambda (x,t) \times \Delta A \times \Delta t \) for small ΔA, Δt.

The first problem is how to treat multiple phases of the same eruptive centre. While we have data on the two most recent examples (Mt Wellington and Rangitoto), geological studies of Searle and others (see Table 2 references) indicate that there were probably multiple phases at several of the other centres, particularly Pupuke, Three Kings, and Mt Eden, for which we have no specific age information. Thus, including the second-events from only two sites (Mt Wellington 2 and Rangitoto 2) in the hazard estimation will bias both the temporal and spatial aspects. The best solution for these incomplete data is, hence, to model only the hazard of a new centre, and consider the ‘re-activation’ of a previous centre separately, assuming that sufficient data can be assembled. Effectively, we suppose that from a geologic perspective, multiple phases from the same eruptive vent constitute the same hazard as a continuous eruption from the vent. Such multiple phase behaviour has been observed for historical eruptions, such as Cerro Negro, over periods of decades, or even longer (Hill et al. 1998).

Magill et al. (2005) model

Previous age models for the AVF were considered unreliable, which motivated Magill et al. (2005) to construct a spatial mixture renewal model, attempting to forecast only the location of the next eruption. This was based on the apparent tendency of eruptions in the AVF to cluster at distances <4,600 m. This approach reduced the reliance on age determination, although the order of the eruptions was all-important. Using the eruption order from Allen and Smith (1994), Magill et al. (2005) showed that the distance between consecutive vents was significantly smaller than that between all pairs of vents, concluding that eruptions do not occur randomly, but rather preferentially closer to the previous eruption. The distance of 4,600 m was then used to define clusters, with an eruption being assigned to the same cluster as the previous event if the distance between the two eruption centres was less than 4,600 m. This collapsed the 49 eruptions into 18 clusters.

In the above construction, the vent locations were justifiably considered known. Hence the regularity distance is unaffected by any errors in dating. Given that the cluster construction process used only the order of the eruptions (and not the ages) from Allen and Smith (1994), they supported a high degree of clustering. However, there appears to have been a tendency in the Allen and Smith (1994) age model to assign vents with undetermined age to a place in the order consistent with their close spatial neighbours (Ian Smith, pers. comm.), and thus there was an unintended circular dependence between clustering and ordering.

With the new age-order model, and applying the dating algorithm above for 1000 Monte-Carlo runs, considerable differences in eruption order emerge in comparison to the Allen and Smith (1994) model. First, if we consider the order given by the simulated mean age in Table 3, we see that the clusters identified by Magill et al. (2005) are no longer as evident (Fig. 4). Further, using the 1000 Monte Carlo run data (not the results in Table 3), the distribution function for the inter-vent distance between successive eruptions for the new age model is no longer significantly different to that between all pairs of centres (Fig. 5).
Fig. 4

Comparison of Allen and Smith (1994) and new eruption order for the AVF

Fig. 5

Distances between successive eruption centres in the AVF. Dashed line is Allen and Smith (1994) order of events, the solid line represents 1000 Monte Carlo repetitions of the new age-order, the dot-dashed line is the result from all possible pairs, and the dotted line is a 95% confidence band based on the latter

Spatio-temporal properties

As the assumptions behind the Magill et al. (2005) model no longer appear valid, a new estimate of the hazard is needed. This requires examination of the spatial and temporal structure of the data, including the distances and azimuths between successive events (Fig. 6). While the distance between all pairs of centres appears to be Weibull distributed (cf. Magill et al. 2005), there seems to be some evidence of more structure in the new Monte Carlo results, though this is influenced by certain order-pairs, such as Purchas Hill—Mt Wellington—Rangitoto, being present in most or all of the simulations. The all-pairs azimuth distribution, including its two preferred directions, appears to be reproduced in the new Monte Carlo simulated order, implying that there is little information about the direction of the next event that can be extracted from the location of the previous event.
Fig. 6

Distances (a, km) and azimuths (b, degrees clockwise from East) between successive eruptions. Top plots show the result from all pairs of centres, whereas the bottom graphs are the result from 1000 Monte Carlo repetitions of the new dating algorithm

The second order properties, i.e., the relationship of distances and azimuths between successive pairs of events, are shown in Fig. 7. The main pattern is a tendency to avoid the same azimuth between two successive pairs of events, and that the previous inter-centre distance provides no information about the next inter-centre distance. Overall, it appears that the location of the previous centre has no effect on the next centre, beyond those effects from the overall spatial structure of the field. Further, Fig. 8 shows that the temporal and spatial distances between successive events appear to be unrelated. This means that a spatio-temporal model with independent temporal and spatial terms is justified. Thus, we can decompose the point process intensity as \( \lambda (x,t) = \lambda (t)f(x) \), and estimate the terms separately.
Fig. 7

Distances (a, km) and azimuths (b, degrees clockwise from East) between pairs of successive eruptions, 1000 Monte Carlo repetitions of the new dating algorithm

Fig. 8

Time versus distance between successive events (based on 1000 Monte Carlo simulations of the new age-order model)

Temporal component of hazard

Figure 9 shows the temporal structure for the AVF, both between successive events and over the life of the field. The distribution of the inter-event times is not memoryless (i.e., a Poisson process), instead exhibiting a tendency to cluster at short time intervals. This obviously corresponds to the peak(s) in the kernel estimate of the eruption rate at about 30 ka. The deviation from Poisson behaviour at longer intervals is a function of the sparse and imprecisely dated early record. The same effect is noticeable on a second order plot of successive inter-event times.
Fig. 9

a Distribution of times between successive events; and b a kernel-smoothed estimate of the time-varying eruption onset rate (Turner et al. 2008a). The dashed line in a is the reference line from an exponential distribution (a Poisson process)

In defining the form of the temporal model component it must first be recognised that the process is clearly not stationary, and thus not a renewal model (Bebbington and Lai 1996). Also, the volume anomaly of Rangitoto (comprising up to 50% of the field volume in one centre; Kermode, 1992) makes a volume-dependent model (Bebbington 2008) infeasible. In addition, because there appears to be less than a full cycle in the intensity, a trend renewal model (Bebbington 2010), or a hidden Markov model approach (Bebbington 2007) is not feasible. However, the clustering at short intervals indicates that a triggering model might be appropriate.

Let us denote the event times as occurring at t1<t2<⋯<tN, where the observation period is \( t \in ({T_1},{T_2}) \), say, and assume that new eruptions occur at random in a Poisson process with a ‘background rate’ μ. Further, suppose that the occurrence of an eruption in the recent past raises the likelihood of a further eruption, with an effect that declines over time. The point process intensity (see, for example, Daley and Vere-Jones 2003) can then be written as
$$ \lambda (t) = \mu + { }\nu \sum\limits_{j:{t_j} < t} {g(t - {t_j})}, $$
(2)
where the probability (conditional on the history of the process) of an event in the time interval (t,tt) will be approximately λ(tt, for small Δt. We require \( \nu \int_0^\infty {g(s)ds < 1} \) to ensure that the process remains stable, i.e., that each event produces less than one additional event on average.
The point process intensity (Eq. 2) is the ‘self-exciting’ model of Hawkes (1971), subsequently applied to earthquakes by Hawkes and Adamopoulos (1973), Vere-Jones and Ozaki (1982), and Ogata (1988). For the excitation function g, physical considerations require that it be monotonically decreasing, and not have an artificial hard cut-off. Hence we will use
$$ g(s) = \frac{1}{\sigma }\exp \left( { - \frac{{{s^2}}}{{2{\sigma^2}}}} \right), $$
the probability density function of a half-normal random variable with mean \( \sigma \sqrt {{2/\pi }} \), which means that the stability condition becomes v < 1. Thus the effect declines in a sigmoid fashion, in contrast to the exponential [\( g(s) = \exp ( - \eta s) \)] and hyperbolic [g(s) = sp] decays used for earthquake aftershock sequences. This difference is driven by the structure of the data; with volcanic data at this resolution we are unable to separate eruptions close together in time, and volcanic eruptions do not occur in accordance with an Omori-type law, where the rate of triggered events declines as the reciprocal of the elapsed time. The parameters μ, ν and σ can be estimated using maximum likelihood, where the loglikelihood is
$$ \log {\hbox{L}} = \sum\limits_{i = 1}^N {\log \lambda ({t_i}) - \int\limits_{{T_1}}^{{T_2}} {\lambda (t)dt} } . $$
An example for 1000 Monte Carlo realizations of the dating algorithm is shown in Fig. 10. The parameter estimates are μ = 0.000059 ± 0.000007/year, ν = 0.71 ± 0.04/year, and σ =3161 ± 711 years. This means that the additional contribution to the rate from a previous event starts at 0.00022/year, and declines to 0.0001/year after 4021 years, and 0.00005/year after 5,479 years. Hence the contribution to the present day hazard from Rangitoto is still considerable (Fig. 10b).
Fig. 10

a Kernel-smoothed estimate of the time-varying eruptive onset intensity (dashed line) and average fitted triggering intensity (solid line) for 1000 Monte Carlo realizations of the new age model. b Forecast intensity (solid line) and 90% confidence bound (dashed lines)

While there are some visual differences between the intensities in Fig. 10a, it must be remembered that while the point intensity is calculated only from the past events, the kernel estimate ‘looks ahead’ to have a contribution from events that occur in the following few thousand years. Also, it is quite possible that some of the poorly constrained events in Table 3 actually occurred in batches, which is not reflected in the Monte Carlo simulation algorithm.

Spatial component of hazard

Spatial effects seem to be from the field as a whole; hence we considered a spatial density kernel (Connor and Connor 2009). The intensity at a point x is estimated as
$$ f(x) = \frac{1}{{2\pi \sqrt {{\left| {\hbox{H}} \right|}} }}\sum\limits_{i = 1}^N {\exp \left( { - \frac{1}{2}{{(x - {x_i})}^{\rm{T}}}{{\hbox{H}}^{ - 1}}(x - {x_i})} \right)} $$
where the bandwidth matrix H is estimated using least squares cross validation (Duong 2007) as
$$ {\hbox{H}} = \left[ {\begin{array}{*{20}{c}} {10.0} & {16.8} \\{16.8} & {30.0} \\\end{array} } \right],\,{\hbox{and}}\,{\hbox{hence}}\,\sqrt {\hbox{H}} = \left[ {\begin{array}{*{20}{c}} {2.1} & {2.4} \\{2.4} & {4.9} \\\end{array} } \right]{\hbox{km}}{.} $$
This indicates east–west and north–south smoothing distances of 4.2 and 9.8 km, respectively, and a clockwise rotation of the kernel. In other words, there is a strong NE-SW spatial structure evident in the AVF (Fig. 11).
Fig. 11

Kernel-smoothed estimate of the spatial intensity of eruption sites in the AVF, probability contours at intervals of 0.001. Triangles represent eruption centres

Discussion and conclusions

The development of the new age model presented here was made possible by the correlation of tephra layers with the AVF eruption centres based on their preserved thicknesses, the known pyroclastic volume of the eruption centres, and their location with respect to the core site. All of these are carried out within the known centre-age constraints. There is the potential for miscorrelations in our approach due to: 1) the potential of reworking and hence possible thickening or thinning of individual tephras in the cores (as commented on by Molloy et al. 2009), compared to an idealised fall thickness; 2) specific wind directions during eruption events producing narrowly distributed and elongate tephra fall thickness attenuations; and 3) that single eruption centres may have produced more than one tephra layer, as recognised for the most recent centres of the field, Mt Wellington and Rangitoto (Searle 1962; Horrocks et al. 2005a). Further work to test and develop this model hence requires a geochemical or petrological approach that extends beyond the problematic broad compositional ranges reported for glass chemistry from the field (e.g., Shane and Smith 2000). Examining component lithologies of tephra particles may also help add another layer of constraint, e.g., Molloy et al. (2009) recognise that AVF6-9 contain large proportions of accidental (non-volcanic) rock fragments, implying deposits from phreatomagmatic eruptions.

Under this new age model, the conclusion of Magill et al. (2005) that there is significant spatio-temporal clustering can be refuted. Instead the data suggest that spatial and temporal eruption recurrences in the AVF are independent of one another. Hence the location of the past event may not shed any light on the location of the next. The age data for events in the field are, however, only strong for the record since 50 ka ago, even with the additional core-tephra information. Older deposits’ ages are still very poorly constrained by radiometric dating; further work is needed to provide information about the earlier development of the volcanic field.

Past workers have described the activity of the field as being ‘spasmodic’ (Cassidy et al. 1999) with episodes of high intensity separated by long pauses. Through geomagnetic work, the propensity for the field to produce rapid sequences of up to five disparate eruptions over periods of less than 100 years was confirmed (Cassata et al. 2008). Independently, the analysis of tephras within sediment cores from the field has also revealed periods of high eruption event intensity, including a major “flare-up” in explosive activity at 32 ± 2 ka (Molloy et al. 2009). Our new age-order analysis clearly shows a strong numerical dominance of known events from the field in the period 40–20 ka (Fig. 10), confirming this ‘flare-up’ behaviour. This period accounts for over 63% of the known eruptions from the AVF, occuring within only 8% of the field’s total life. Before this period, there is a period at least 160 ka long over which 10–12 eruptions are distributed, with the limited information available suggesting that many of them occurred concurrently at the inception of the field. The latest 20 ka includes up to 8 eruptions from only six eruptive centres, including the latest event that erupted a volume of magma equivalent to that produced collectively by all eruptions that had occurred before (Allen and Smith 1994). Based on this highly variable history, it can be assumed that there is no ‘steady-state’ approximation that can be applied to the eruption intensity or hazard at the AVF. Instead, its behaviour appears to have been controlled by a few times during which magma-supply rates were sharply higher, with the latest high-supply event being focused in a single eruption, and concentrated at a single point source.

Our results suggest that the spatial hazard intensity, or the spatial probability of vent opening, is strongly structurally controlled with a NE-SW orientation that is parallel to the strike of the subduction zone c. 500 km to the east, and to the orientation of the Taupo Volcanic Zone, dominating the Central North Island to the south (Fig. 1). This orientation is also sub-parallel to many North Island fault systems of the region to the south and east (Edbrooke 2001). The trend is perpendicular to regional magnetic and gravity anomalies of c. 340° to 325° representing the crustal structure (Spörli and Eastwood 1997), and may relate to their proposal that the AVF lies within a releasing structure of a dextral strike-slip regime.

In a global context, the AVF is a very small field with a relatively short geological history. In longer lived volcanic field systems, eruption intensity is also recognised to vary considerably over time, with pulses of high intensity much shorter than the million-year timescales of field life (e.g., Condit and Connor 1996). In these longer-lived systems, however, dating resolution does not allow the degree of analysis afforded at Auckland. Hence, the AVF data may represent a window onto a “pulsing” behaviour of major longer-lived volcanic fields, and shows that pulsatory magma supply on a range of temporal scales appears to characterise the behaviour of monogenetic volcanic fields.

Notes

Acknowledgements

We wish to acknowledge support by the NZ FRST-IIOF Grant “Facing the challenge of Auckland’s Volcanism” (MAUX0808). We thank Jan Lindsay and Ian Smith (U of Auckland) for valuable discussion on event ages and other features of the AVF and Kate Arentsen (Massey U) for comments on the manuscript. Reviews by Olivier Jaquet and an anonymous referee led to important improvements in the paper.

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Authors and Affiliations

  1. 1.Volcanic Risk SolutionsMassey UniversityPalmerston NorthNew Zealand
  2. 2.Institute of Fundamental Sciences—StatisticsMassey UniversityPalmerston NorthNew Zealand

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